In recent years, various operators of fractional calculus (that is, calculus of integrals and derivatives of arbitrary real or complex orders) have been investigated and applied in many remarkably diverse fields of science and engineering. Many authors have demonstrated the usefulness of fractional calculus in the derivation of particular solutions of a number of linear ordinary and partial differential equations of the second and higher orders. The purpose of this paper is to present a certain class of the explicit particular solutions of the associated Cauchy-Euler fractional partial differential equation of arbitrary real or complex orders and their applications as follows:
MSC: 26A33, 33C10, 34A05.
Keywords:fractional calculus; differential equation; partial differential equation; generalized Leibniz rule; analytic function; index law; linearity property; principle value; initial and boundary value
Dedicated to Professor Hari M Srivastava.
1 Introduction, definitions and preliminaries
The subject of fractional calculus (that is, derivatives and integrals of any real or complex order) has gained importance and popularity during the past two decades or so, due mainly to its demonstrated applications in numerous seemingly diverse fields of science and engineering (cf.[1-15]). By applying the following definition of a fractional differential (that is, fractional derivative and fractional integral) of order , many authors have obtained particular solutions of a number of families of homogeneous (as well as nonhomogeneous) linear fractional differ-integral equations.
In this paper, we present a direct way to obtain explicit solutions of such types of the associated Cauchy-Euler fractional partial differential equation with initial and boundary values. The results are a coincidence that the solutions are obtained by the methods applying the Laplace transform with the residue theorem. In this paper, we present some useful definitions and preliminaries for the paper as follows.
First of all, we find it is worthwhile to recall here the following useful lemmas and properties associated with the fractional differ-integration which is defined above.
Lemma 1.1 (Linearity property)
Lemma 1.2 (Index law)
Lemma 1.3 (Generalized Leibniz rule)
Lemma 1.4 (Cauchy’s residue theorem)
For a constanta,
Proof The proofs between ‘ν is not an integer’ and ‘ν is an integer’ are not coincident, so we mention the proof as follows.
Property 1.2For a constanta,
Property 1.3For a constanta,
wheresis the Laplace complex parameter. We recall from the fundamental formula (cf. )
2 Main results
Theorem 2.1The fractional partial differential equation
So that the given equation (2.1) becomes
Equation (2.4) leads to the auxiliary equation
There are three different cases to be considered, depending on whether the roots of this quadratic equation (2.5) are distinct real roots, equal real roots (repeated real roots), or complex roots (roots appear as a conjugate pair). The three cases are due to the discriminant of the coefficients .
Theorem 2.2The fractional partial differential equation
Proof The similarity between the forms of solutions of Equation (2.1) and solutions of a linear equation with constant coefficients of Equation (2.6) is not just a coincidence.
So that the given equation (2.6) becomes
The analysis of three cases is similar to Theorem 2.1, we can obtain each solution of the forms as follows:
Remark The constant λ in Equations (2.2) and (2.7) can be solved directly by constant initial value and constant boundary values (or by the numerical methods).
Corollary 2.1The fractional partial differential equation
Corollary 2.2The fractional partial differential equation
then it has solutions of the form
Solution Equation (3.1) is coincident to
We have the solution
Example 3.2 The fractional partial differential equation
Example 3.3 The fractional partial differential equation
Example 3.4 The fractional partial differential equation
and the particular solution is
The solution obtained by the method of Laplace transform and the residue theorem is a coincidence, which is our result above. □
The authors declare that they have no competing interests.
SDL carried out the molecular genetic studies, participated in the sequence alignment and drafted the manuscript. CHL carried out the immunoassays. SMS participated in the sequence alignment.
The authors are deeply appreciative of the comments and suggestions offered by the referees for improving the quality and rigor of this paper. The present investigation was supported, in part, by the National Science Council of the Republic of China under Grant NSC-101-2115-M-033-002.
Lin, S-D, Chang, L-F, Srivastava, HM: Some families of series identities and associated fractional differintegral formulas . Integral Transforms Spec. Funct.. 20(10), 737–749 (2009). Publisher Full Text
Srivastava, HM, Owa, S, Nishimoto, K: Some fractional differintegral equations . J. Math. Anal. Appl.. 106, 360–366 (1985). Publisher Full Text
Tu, S-T, Chyan, D-K, Srivastava, HM: Some families of ordinary and partial fractional differintegral equations . Integral Transforms Spec. Funct.. 11, 291–302 (2001). Publisher Full Text
Eidelman, SD, Kochubei, AN: Cauchy problem for fractional diffusion equation . J. Differ. Equ.. 199(2), 211–255 (2004). Publisher Full Text
Schneider, WR, Wyss, W: Fractional diffusion and wave equation . J. Math. Phys.. 30(1), 134–144 (1989). Publisher Full Text
Mainardi, F: The fundamental solutions for the fractional diffusion-wave equation . Appl. Math. Lett.. 9(6), 23–28 (1996). Publisher Full Text